Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 916741, 10 pages

http://dx.doi.org/10.1155/2015/916741

## A New Algorithm for Reconstructing Two-Dimensional Temperature Distribution by Ultrasonic Thermometry

^{1}School of Automation, Chongqing University, Chongqing 400044, China^{2}Key Laboratory of Dependable Service Computing in Cyber Physical Society, MOE, Chongqing 400044, China^{3}School of Software Engineering, Chongqing University, Chongqing 400044, China

Received 8 August 2014; Accepted 29 January 2015

Academic Editor: Muhammad N. Akram

Copyright © 2015 Xuehua Shen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Temperature, especially temperature distribution, is one of the most fundamental and vital parameters for theoretical study and control of various industrial applications. In this paper, ultrasonic thermometry to reconstruct temperature distribution is investigated, referring to the dependence of ultrasound velocity on temperature. In practical applications of this ultrasonic technique, reconstruction algorithm based on least square method is commonly used. However, it has a limitation that the amount of divided blocks of measure area cannot exceed the amount of effective travel paths, which eventually leads to its inability to offer sufficient temperature information. To make up for this defect, an improved reconstruction algorithm based on least square method and multiquadric interpolation is presented. And then, its reconstruction performance is validated via numerical studies using four temperature distribution models with different complexity and is compared with that of algorithm based on least square method. Comparison and analysis indicate that the algorithm presented in this paper has more excellent reconstruction performance, as the reconstructed temperature distributions will not lose information near the edge of area while with small errors, and its mean reconstruction time is short enough that can meet the real-time demand.

#### 1. Introduction

Temperature, especially temperature distribution is one of the most fundamental and vital parameters for theoretical study and control of various industrial applications, particularly of which related to burning and heating. Typical examples can be seen from applications of industrial furnaces. Temperature information is essential for the development of combustion theory and optimization of combustion systems, since every part of a combustion event, such as ignition, burnout, and evolution of emissions, is related to temperature [1, 2]. Information about temperature distribution is helpful for detecting and correcting hot spots which may cause safety accidents like explosions and spontaneous ignitions; meanwhile, it also contributes to designing combustion systems of low , as presence of hot spots always increases generation [3, 4]. Furthermore, temperature distribution always immediately influences the combustion efficiency of pulverized coal, the structure and state of reaction, and the safety of operation [4–6]. Apart from these applications of industrial furnaces, temperature distribution and its reconstruction during combustion are of great interest in automotive and aerospace industries as well, because engine properties need to be monitored via studying flame dynamics [7, 8]. In particular, as flames in high-temperature combustions region generally become invisible, special and effective measurement techniques are required greatly to identify the combustion state [9]. It can be concluded from the above examples that scientific research, economic benefits, ecological environment, and security requirements have aroused a strong need for improving thermometry techniques in industry fields.

Temperature measurement techniques of thermocouples and thermal resistances have been applied to industry for a long time. However, as intrusive methods, they require instruments to direct contact with measure medium, and the relatively slow time response in measurement always brings about inaccuracy; simultaneously, the temperature information gathered is insufficient owing to these methods’ single-point measurement [2, 10–12]. The infrared radiation technique is convenient but just for measuring surface temperature, and influences due to different emissivity and reflection of infrared radiations from other sources often result in deterioration of accuracy [11–13]. Although methods using fiber optics become popular recently, they are not suitable for monitoring non-uniform temperature distribution either. For one thing, most of fiber optics also need direct contact with measure medium, which is similar to thermocouples and thermal resistances [13, 14]; for another, this technique is always accompanied with a narrow operating range from 0 to 330°C, while temperature in applications is always beyond this range [15].

Some studies of temperature distribution reconstruction have emerged in previous researches. The method based on analyzing colored radiation images captured by CCD cameras is working by a model relating the radiation image with temperature distribution [2, 16]. However, it is suitable only for visible luminous combustions or flames and not available for invisible situations. An inverse analysis has been presented for reconstruction of temperature profile in flames [17]. But it is a pity that since the cross section of flame is divided into a series of concentric rings and temperature in each ring is expressed as a discrete value, it is just effective for axisymmetric free flames. A laser ultrasonic method for evaluation of temperature distribution is presented on the basis of temperature dependence of ultrasound [10, 11]. Nevertheless, it is practicable just for surface temperature distribution, as the surface acoustic waves used here propagate merely on the surface of medium. The CT method for calculating temperature distribution is also an effective reconstruction means of ultrasonic technique [18]. Unfortunately, it requires large-scale machinery and a large number of transducers to collect enough projection data; thus it is bound to increase cost and running time. As another acoustic method, temperature distribution reconstruction using algorithm based on RBF neural network shows great advantages when compared to that using algorithm based on least square method [19]. However, it requires a lot of training samples to calculate appropriate parameters of the neural network, while training samples are almost impossible to obtain in practice, even if obtained they are not accurate enough.

To overcome the shortcomings of conventional temperature measurement techniques and make up for the deficiencies of existing temperature distribution reconstruction methods, a nonintrusive ultrasonic technique of temperature distribution reconstruction is investigated in this paper, which is also presented based on the dependence of ultrasound velocity on temperature [3, 10–12, 19–23]. Temperature distribution reconstruction by this ultrasonic thermometry is an inversion problem, whose solution depends greatly on reconstruction algorithm. In practical application of this technique, reconstruction algorithm based on least square method has been commonly used, because it is simplest with high stability and can gain best match data by minimizing the error square sum. However, the temperature information it estimated is too limited to offer an accurate reflection of temperature distribution. Aiming at this shortage, an improved reconstruction algorithm based on least square method and multiquadric interpolation is presented, as it can deal with sparse data well while being accompanied with small error.

This paper is organized as follows. Section 2 describes the basic principle of ultrasonic thermometry. Section 3 presents a reconstruction algorithm based on least square method and multiquadric interpolation. Section 4 utilizes four two-dimensional temperature distribution models with different complexity in numerical studies to validate the reconstruction performance of our presented algorithm and gives experiment results and analysis. Conclusions and future research are presented in Section 5.

#### 2. The Principle of Ultrasonic Thermometry

##### 2.1. The Relationship between Ultrasound Velocity and Temperature

Ultrasonic thermometry is based on the dependence of ultrasound velocity on temperature, or that ultrasound velocity in any medium is a function of temperature. Such that, in ideal gases, the ultrasound velocity is directly proportional to the square root of temperature, and in most of liquids the dependence is linear, while in solid objects the ultrasound velocity generally decreases with the increment of temperature [22]. Considering most widely situations in practice, principle based on ideal gas is given in detail.

The relationship between ultrasound velocity and temperature can be described as the following equation [3, 10–12, 19–22]: where is the ultrasound velocity, is the gas absolute temperature, is the universal gas constant, and and are the ratio and the average molecular weight of the gas mixture, respectively. As , , and are fixed constants that can be measured for the specific gas, they may be replaced by a coefficient which is also a constant.

When the ultrasound velocity and gas properties are known, temperature can be calculated by (2) derived from (1) as follows:

A simplest ultrasonic thermometry instrument is composed by one ultrasonic transmitter mounted on one side and one ultrasonic receiver mounted on the other side along the same path. The transmitter is used to produce an ultrasonic signal at a specific time, and the receiver at a known distance is responsible for detecting the ultrasonic signal. As the distance between ultrasonic transmitter and ultrasonic receiver is fixed and known, ultrasound velocity can be calculated when travel time is measured. Thus (2) can be rewritten as where is the distance between ultrasonic transmitter and ultrasonic receiver and is the total travel time that the ultrasound is in transit. From (3), average temperature along the path is obtained.

##### 2.2. Temperature Distribution Reconstruction

Temperature distribution reconstruction will be accomplished using multiple ultrasonic transmitters and receivers (or collectively called ultrasonic transducers) installed over the measure area, as paths between them can be combined to compute temperature distribution with suitable algorithms [3, 19–21]. With multiple transducers installed, the measure areas should be divided into several blocks according to specific situations. Figure 1 shows the typical situations in practical applications, which are square and circular measure areas, respectively. In the figures, transducers are represented by the black symbols, effective paths are represented by the solid lines within the areas, and measure areas are divided into blocks by the dashed lines.